Without performing division, determine whether the decimal expansion of is terminating or non-terminating. If it terminates, state the number of decimal places.
The decimal expansion of 18/125 is terminating.
To determine this, we find the prime factorisation of the denominator 125, which is 5³. Since the prime factors of the denominator are only 5, the decimal expansion will be terminating.
To find the number of decimal places, we make the denominator a power of 10 by multiplying both numerator and denominator by 2³ (which is 8). So, 18/125 = () / () = 144 / 1000 = 0.144. Since the denominator is 10³, the decimal terminates at 3 decimal places.
Explanation
The textbook context explains that the decimal expansion of a rational number p/q terminates precisely when the prime factors of q are only 2, only 5, or both. For 18/125, the denominator 125 has the prime factorisation 5³, satisfying this condition. To find the number of decimal places, we convert the denominator to a power of 10 by multiplying by 2³, resulting in 1000 (or 10³), which indicates exactly 3 decimal places.
Solution Steps
Step 1: Prime factorisation of denominator: 125 = 5³. Since the prime factors are only 5, the decimal expansion is terminating.
Step 2: Convert to power of 10: Multiply numerator and denominator by 2³ (i.e., 8) to get a denominator of 10³. () / () = 144 / 1000.
Step 3: Determine decimal places: Since the denominator is 10³, the decimal expansion has 3 decimal places (0.144).